Page 232 - Handbook of Civil Engineering Calculations, Second Edition

REINFORCED CONCRETE 2.17

FIGURE 9. Analysis of two-way slab by mechanism method.

Calculation Procedure:

1. Draw line GH perpendicular to AE at E; express distances b
and c in terms of a
Consider a slab to be reinforced in orthogonal directions. If the reinforcement in one
direction is identical with that in the other direction, the slab is said to be isotropically
reinforced; if the reinforcements differ, the slab is described as orthogonally anisotropic.
In the former case, the capacity of the slab is identical in all directions; in the latter case,
the capacity has a unique value in every direction. In this instance, assume that the slab
size is excessive with respect to balanced design, the result being that the failure of the
slab will be characterized by yielding of the steel.
In a steel beam, a plastic hinge forms at a section; in a slab, a plastic hinge is assumed
to form along a straight line, termed a yield line. It is plausible to assume that by virtue of
symmetry of loading and support conditions the slab in Fig. 9a will fail by the formation of
a central yield line EF and diagonal yield lines such as AE, the ultimate moment at these
lines being positive. The ultimate unit moment m u is the moment acting on a unit length.
Although it is possible to derive equations that give the location of the yield lines, this
procedure is not feasible because the resulting equations would be unduly cumbersome.
The procedure followed in practice is to assign a group of values to the distance a and to
determine the corresponding values of m u . The true value of m u is the highest one ob-
tained. Either the static or mechanism method of analysis may be applied; the latter will
be applied here.
Expressing the distances b and c in terms of a gives tan a 6/a AE/b c/(AE); b
aAE/6; c 6AE/a.
2. Find the rotation of the plastic hinges
Allow line EF to undergo a virtual displacement after the collapse load is reached.
During the virtual displacement, the portions of the slab bounded by the yield lines and